Nearly Optimal Algorithms for Canonical Matrix Forms

A Las Vegas type probabilistic algorithm is presented for nding the Frobenius canonical form of an n n matrix T over any eld K. The algorithm requires O~(MM(n)) = MM(n) (logn) O(1) operations in K, where O(MM(n)) operations in K are suucient to multiply two n n matrices over K. This nearly matches the lower bound of (MM(n)) operations in K for this problem, and improves on the O(n 4) operations in K required by the previously best known algorithms. We also demonstrate a fast parallel implementation of our algorithm for the Frobenius form, which is processor-eecient on a PRAM. As an application we give an algorithm to evaluate a polynomial g 2 Kx] at T which requires only O~(MM(n)) operations in K when degg n 2. Other applications include sequential and parallel algorithms for computing the minimal and characteristic polynomials of a matrix, the rational Jordan form of a matrix, for testing whether two matrices are similar, and for matrix powering, which are substantially faster than those previously known. Computing a canonical or normal form of an n n matrix T over any eld K is a classical mathematical problem with many practical applications. A fundamental theorem of linear algebra states that any T 2 K nn is similar to a unique matrix S 2 K nn of the block diagonal form